Model checking quantum Markov chains
Model checking quantum Markov chains Yuan Feng, Nengkun Yu, and Mingsheng Ying University of Technology Sydney, Australia, Tsinghua University, China Model checking quantum Markov chains. Journal of Computer and System Sciences 79, 1181-1198, (2013) Reachability of recursive quantum Markov chains. Proceedings of the 38th Int. Symp. on Mathematical Foundations of Computer Science (MFCS’13) 385-396.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Motivation
Quantum mechanics is highly counterintuitive; flaws and errors creep in during the design of quantum programs and quantum protocols. So, it is indispensable to develop techniques of verifying and debugging quantum systems.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Model checking
Model-checking is one of the dominant techniques for verification of classical hardware as well as software systems. It has proved mature as witnessed by a large number of successful industrial applications. Quantum model checking???
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Probability Theory v.s. Quantum Information Theory
Quantum bit: Binary Random Varable X: X = 0 or X = 1 1 0
Unit vector in a 2D Hilbert space |φi = a0 |0i + a1 |1i, ai ∈ C , |a0 |2 + |a1 |2 = 1
|1i
|φi |0i
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Probability Theory v.s. Quantum Information Theory
Evolution:
Stochastic Matrices
1 2 1 2
1 2 1 2
p0 p1
Unitary Matrices
Preserve l2 -norm |φ0 i = U · |φi
Preserve l1 -norm p0 = S · p
Evolution:
=
1 2 1 2
√1 2 √1 2
√1 2 − √12
!
a0 a1
=
√1 (a0 2 √1 (a0 2
+ a1 ) − a1 )
!
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Probability Theory v.s. Quantum Information Theory
Measurement:
Observation: Pr(X = b ) = pb , pb ∈ [0, 1]
A measurement of |φi according to a Hermitian operator M = ∑i λi |bi ihbi | is a projection onto the orthonormal vectors |bi i, and Pr[outcome is λi ] = |hφ|bi i|2 .
|1i |φi⊥
|φi |0i
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Density operators
Mixed state: Classical distribution over (pure) quantum states. |φ1 i, with probability p1 . . . . ρ= . . |φk i, with probability pk Ensemble: {pi : |φi i}. Density operator: ρ = ∑ki=1 pi |φi ihφi | (hermitian, trace 1, positive) Contains all information about the state. Different ensembles can have the same density operator.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Density operators
Different ensembles can have the same density operator. ( 1 √1 (|0i − |1i), w.p. 2 2 = |0i, w.p. 12 √3 1 1 3 2 |0i − 2 |1i, w.p. √3 1 − 3 1 4 4 |0i, w.p. 4 (1 − √3 ) = − 14 14 |1i, w.p. 14 (1 − √13 )
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Super-operators and Kraus theorem Super-operators: (special) mapping from density operators to density operators. Kraus representation theorem: super-operator if and only if
A map E is a
d
E (ρ) =
∑ Ei ρEi†
i =1
for some set of matrices {Ei , i = 1, . . . , d } with ∑i Ei† Ei ≤ I . Special case: Unitary transformation: ρ → UρU † Measurement with outcome i: ρ → |bi ihbi |ρ|bi ihbi | Measurement with reading outcome: ρ → ∑i |bi ihbi |ρ|bi ihbi |
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Matrix representation of super-operators
Let E = {Ei : i ∈ I } be a super-operator. The matrix representation of E is defined as ME =
∑ Ei ⊗ Ei∗ .
i ∈I
Here the complex conjugate is taken according to the orthonormal basis {|k i : k ∈ K }. It is easy to check that ME is independent of the choice of orthonormal basis and the Kraus operators Ei .
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Markov chains
A Markov chain (MC) is a tuple (S, P ) where S is a countable set of states; P : S × S → [0, 1] such that for each s ∈ S,
∑ P (s, t ) = 1,
t ∈S
or equivalently, P (s, ·) is a probabilistic distribution over S.
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Quantum Markov chains
(S, P )
⇒
Set S
⇒
Prob. distributions
⇒
P : Dist (S ) → Dist (S )
⇒
(H, E ) Hilbert space H Density operators
E : D(H) → D(H)
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Obstacles for model checking quantum system
The set of all possible quantum states, H, is a continuum, even when it is finite dimensional. The techniques of classical model checking, which normally work for finite state spaces, cannot be applied directly.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
In this talk, we propose...
A super-operator weighted Markov chain model which aims at providing finite models for general quantum programs and quantum communication protocols. A quantum extension QCTL of the logic PCTL to descibe properties we are interested in for QMCs. An algorithm to model check logic formulas in QCTL against a QMC model.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Some more notations
Let SO(H) be the set of super-operators on H, ranged over by E, F, · · · . Definition Let E , F ∈ SO(H). 1
2
E v F if for any ρ ∈ D(H), F (ρ) − E (ρ) is positive semi-definite; E . F if for any ρ ∈ D(H), tr(E (ρ)) ≤ tr(F (ρ)).
Let h be . ∩ &; it is obviously an equivalence relation.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Some notations
Let
SI(H) = {E ∈ SO(H) : E . IH } be the ‘quantum’ correspondence of the unit interval [0, 1] for real numbers.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Quantum Markov chains
A super-operator weighted Markov chain, or quantum Markov chain (QMC), over H is a tuple (S, Q, AP, L), where S is a countable set of states;
Q : S × S → SI(H) such that for each s ∈ S, ∑t ∈S Q(s, t ) h IH , AP is a finite set of atomic propositions; L is a mapping from S to 2AP . A classical Markov chain may be viewed as a degenerate quantum Markov chain in which all super-operators appear in the transition matrix have the form p IH for some 0 ≤ p ≤ 1.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Example: quantum loop
A simple quantum loop program goes as follows: l0 : q := F (q )
l1 : while M [q ] do
l2 : l3 : od where M = λ0 |0ih0| + λ1 |1ih1|.
q : = E (q )
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Example: quantum loop
l0 Fq Eq
l2
l1 Eq0 Eq1
I
Here Eq0 = {|0iq h0|} and Eq1 = {|1iq h1|}.
l3
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
QCTL
The syntax of quantum computation tree logic (QCTL) is as follows: Φ ::=
a | ¬Φ | Φ ∧ Ψ | Q∼E [ψ]
ψ ::= XΦ | ΦUΨ
where a is an atomic proposition, ∼ ∈ {., &}, and E ∈ SI(H). We call Φ a state formula and ψ a path formula.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
QCTL Let M = (S, Q, AP, L). The satisfaction relation |= is defined inductively: for any state s ∈ S, s |= a iff a ∈ L(s )
s |= ¬Φ iff s 6|= Φ
s |= Φ ∧ Ψ iff s |= Φ and s |= Ψ and for any path π ∈ PathM (s ), π |= XΦ
π |= ΦUΨ
iff iff
π (1) |= Φ
∃i ∈ N.(π (i ) |= Ψ ∧ ∀j < i.(π (j ) |= Φ)).
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
QCTL
Finally, s |= Q∼E [ψ] iff Q M (s, ψ) ∼ E where Q M (s, ψ) = Qs ({π ∈ PathM (s ) | π |= ψ}). But how to define Qs ?
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Super-operator valued measures
Let (Ω, Σ) be a measurable space; that is, Ω is a non-empty set and Σ a σ-algebra over Ω. A function ∆ : Σ → SI(H) is said to be a super-operator valued measure (SVM for short) if ∆ satisfies the following properties: 1 2
∆(Ω) h IH ;
∆( i Ai ) h ∑i ∆(Ai ) for all pairwise disjoint and countable sequence A1 , A2 , . . . in Ω. U
We call the triple (Ω, Σ, ∆) a (super-operator valued) measure space.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Properties of super-operator valued measures Let (Ω, Σ, ∆) be a measure space. Then 1 2 3 4
∆ ( ∅ ) = 0H ;
∆(Ac ) + ∆(A) h IH ;
for any A, A0 ∈ Σ, if A ⊆ A0 then ∆(A) . ∆(A0 ); for any sequence A1 , A2 , . . . in Σ, if A1 ⊆ A2 ⊆ . . . , then there exists a sequence E1 v E2 v . . . in SI(H) such that for any i, S ∆(Ai ) h Ei , and ∆( i ≥1 Ai ) = limi →∞ Ei . if A1 ⊇ A2 ⊇ . . . , then there exists a sequence E1 w E2 w . . . in SI(H) such that for any i, T ∆(Ai ) h Ei , and ∆( i ≥1 Ai ) = limi →∞ Ei .
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
SVM for a QMC Fix a state s ∈ S.
Sample space Ω = PathM (s ).
b ) ⊆ PathM (s ) be defined Let the cylinder set Cyl (π as b ) = {π ∈ PathM (s ) : π b is a prefix of π }; Cyl (π that is, the set of all infinite paths with b prefix π. σ-algebra over Ω: M b) : π b ∈ Pathfin Σs = σ ({Cyl (π (s )}
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
SVM for QMCs
M (s ), we b = s0 . . . sn ∈ Pathfin For any finite path π define the super-operator IH , if n = 0; b) = Q( π Q(sn−1 , sn ) · · · Q(s0 , s1 ), otherwise.
Let a mapping Qs be defined by letting Qs (∅) = 0H and b )) = Q(π b ). Qs (Cyl (π (1)
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Extend Qs to a SVM
Theorem The mapping Qs can be extended to a SVM on the σ-algebra Σs . Furthermore, this extension is unique up to the equivalence relation h. Remark: The main tool we use to prove this theorem is the Kluvanek’s generalisation of the Carath´eodory-Hahn extension theorem from vector measure theory.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
QCTL
Theorem For each path formula ψ and each state s in a QMC M, the set
{π ∈ PathM (s ) | π |= ψ} is measurable.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Back to the example l0 Fq Eq
l2
l1 Eq0 Eq1
I
l3
Let ♦Ψ ≡ ttUΨ. The QCTL formula Q&E [♦ l3 ] asserts that the probability that the loop program terminates is lower bounded by E . That is, for any initial quantum state ρ, the termination probability is not less than tr(E (ρ)). In particular, the property that it terminates everywhere can be described as Q&IH [♦ l3 ].
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Model checking
Given a state s in a qMC M = (S, Q, AP, L) and a state formula Φ expressed in QCTL, model checking if s |= Φ is essentially to determine whether s belongs to the satisfaction set Sat (Φ) = {s ∈ S : s |= Φ} which is defined inductively as follows: Sat (a) = {s ∈ S : a ∈ L(s )}
Sat (¬Ψ) = S \Sat (Ψ)
Sat (Ψ ∧ Φ) = Sat (Ψ) ∩ Sat (Φ)
Sat (Q∼E [ψ]) = {s ∈ S : Q M (s, ψ) ∼ E }. Recall:
Q M (s, ψ) = Qs ({π ∈ PathM (s ) | π |= ψ})
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Case 1: ψ = XΦ
By definition, {π ∈ PathM (s ) : π |= XΦ} = Thus Q M (s, XΦ) = Qs
=
∑
]
Cyl (st ) h
t ∈Sat (Φ)
U
t ∈Sat (Φ)
∑
Cyl (st ).
Qs (Cyl (st ))
t ∈Sat (Φ)
Q(s, t ).
t ∈Sat (Φ)
This can be calculated easily since by the recursive nature of the definition, we can assume that Sat (Φ) is already known.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Case 2: ψ = ΦUΨ
In this case, after some calculation, we get the I , H 0H , Q M (s, ΦUΨ) h ∑ Q M (t, ΦUΨ)Q(s, t ), t ∈S
equation system if s ∈ Sat (Ψ); if s 6∈ Sat (Φ) ∪ Sat (Ψ); if s ∈ Sat (Φ)\Sat (Ψ).
Then for each s ∈ Sat (Φ)\Sat (Ψ), Q M (s, ΦUΨ) h
∑
t ∈Sat (Φ)\Sat (Ψ)
Q M (t, ΦUΨ)Q(s, t ) +
∑
t ∈Sat (Ψ)
Q(s, t ).
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Let S 0 = Sat (Φ)\Sat (Ψ). For any s ∈ S 0 , Q M (s, ΦUΨ) h
∑0 Q M (t, ΦUΨ)Q(s, t ) + ∑
Q(s, t ).
t ∈Sat (Ψ)
t ∈S
Let
T = [Q(t, s )]s,t ∈S 0 and
"
G=
∑
# Q(s, t )
t ∈Sat (Ψ)
. s ∈S 0
Then the required row vector (Q M (s, ΦUΨ))s ∈S 0 is equivalent to the fixed point of the function f (X ) = X T + G .
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
A theorem
Theorem Let f (X ) = X T + G be defined above. Then 1
2
0
f (X ) has the least fixed point, denoted by E 0 , in SI(H)|S | under the order v; Given any E ∈ SI(H) and 1 ≤ i ≤ |S 0 |, it can be decided whether E ∼ Ei0 , ∼ ∈{., &}, in time O (n2 d 4 ) where d = dim (H) is the dimension of H and n = |S 0 |.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Back to the example again We check the property Q&E [♦ l3 ] = Q&E [ttUl3 ] when F = {|+ihi | : i = 0, 1}, E i = {|i ihi |}, i = 0, 1, and E = X . l0 Fq Eq
l2
l1 Eq0 Eq1
I
l3
We first calculate that Sat (l3 ) = {l3 } and Sat (tt) = {l0 , l1 , l2 , l3 }.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Back to the example again
l0 Fq Eq
l2
l1 Eq0 Eq1
I
l3
Q M (l0 , ♦ l3 ) = Q M (l1 , ♦ l3 )F Q M (l1 , ♦ l3 ) = Q M (l2 , ♦ l3 )E 1 + E 0 Q M (l2 , ♦ l3 ) = Q M (l1 , ♦ l3 )E
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Example
We calculate that for i = 0, 1, 2, Q M (li , ♦ l3 ) = Set 0 where Set 0 = {|0ih0|, |0ih1|} h I , and so li |= Q&E [♦ l3 ] for any E . I .
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Summary
A super-operator weighted Markov chain model which aims at providing finite models for general quantum programs and quantum communication protocols. A quantum extension QCTL of the logic PCTL to descibe properties we are interested in for QMCs. An algorithm to model check logic formulas in QCTL against a QMC model.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Topics for further studies
Tools to implement the model checking algorithm. Model checking quantum properties. Check security of physically implemented quantum cryptographic systems.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Thank you! Questions or Comments?
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Motivation
Quantum mechanics is highly counterintuitive; flaws and errors creep in during the design of quantum programs and quantum protocols. So, it is indispensable to develop techniques of verifying and debugging quantum systems.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Model checking
Model-checking is one of the dominant techniques for verification of classical hardware as well as software systems. It has proved mature as witnessed by a large number of successful industrial applications. Quantum model checking???
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Probability Theory v.s. Quantum Information Theory
Quantum bit: Binary Random Varable X: X = 0 or X = 1 1 0
Unit vector in a 2D Hilbert space |φi = a0 |0i + a1 |1i, ai ∈ C , |a0 |2 + |a1 |2 = 1
|1i
|φi |0i
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Probability Theory v.s. Quantum Information Theory
Evolution:
Stochastic Matrices
1 2 1 2
1 2 1 2
p0 p1
Unitary Matrices
Preserve l2 -norm |φ0 i = U · |φi
Preserve l1 -norm p0 = S · p
Evolution:
=
1 2 1 2
√1 2 √1 2
√1 2 − √12
!
a0 a1
=
√1 (a0 2 √1 (a0 2
+ a1 ) − a1 )
!
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Probability Theory v.s. Quantum Information Theory
Measurement:
Observation: Pr(X = b ) = pb , pb ∈ [0, 1]
A measurement of |φi according to a Hermitian operator M = ∑i λi |bi ihbi | is a projection onto the orthonormal vectors |bi i, and Pr[outcome is λi ] = |hφ|bi i|2 .
|1i |φi⊥
|φi |0i
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Density operators
Mixed state: Classical distribution over (pure) quantum states. |φ1 i, with probability p1 . . . . ρ= . . |φk i, with probability pk Ensemble: {pi : |φi i}. Density operator: ρ = ∑ki=1 pi |φi ihφi | (hermitian, trace 1, positive) Contains all information about the state. Different ensembles can have the same density operator.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Density operators
Different ensembles can have the same density operator. ( 1 √1 (|0i − |1i), w.p. 2 2 = |0i, w.p. 12 √3 1 1 3 2 |0i − 2 |1i, w.p. √3 1 − 3 1 4 4 |0i, w.p. 4 (1 − √3 ) = − 14 14 |1i, w.p. 14 (1 − √13 )
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Super-operators and Kraus theorem Super-operators: (special) mapping from density operators to density operators. Kraus representation theorem: super-operator if and only if
A map E is a
d
E (ρ) =
∑ Ei ρEi†
i =1
for some set of matrices {Ei , i = 1, . . . , d } with ∑i Ei† Ei ≤ I . Special case: Unitary transformation: ρ → UρU † Measurement with outcome i: ρ → |bi ihbi |ρ|bi ihbi | Measurement with reading outcome: ρ → ∑i |bi ihbi |ρ|bi ihbi |
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Matrix representation of super-operators
Let E = {Ei : i ∈ I } be a super-operator. The matrix representation of E is defined as ME =
∑ Ei ⊗ Ei∗ .
i ∈I
Here the complex conjugate is taken according to the orthonormal basis {|k i : k ∈ K }. It is easy to check that ME is independent of the choice of orthonormal basis and the Kraus operators Ei .
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Markov chains
A Markov chain (MC) is a tuple (S, P ) where S is a countable set of states; P : S × S → [0, 1] such that for each s ∈ S,
∑ P (s, t ) = 1,
t ∈S
or equivalently, P (s, ·) is a probabilistic distribution over S.
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Quantum Markov chains
(S, P )
⇒
Set S
⇒
Prob. distributions
⇒
P : Dist (S ) → Dist (S )
⇒
(H, E ) Hilbert space H Density operators
E : D(H) → D(H)
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Obstacles for model checking quantum system
The set of all possible quantum states, H, is a continuum, even when it is finite dimensional. The techniques of classical model checking, which normally work for finite state spaces, cannot be applied directly.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
In this talk, we propose...
A super-operator weighted Markov chain model which aims at providing finite models for general quantum programs and quantum communication protocols. A quantum extension QCTL of the logic PCTL to descibe properties we are interested in for QMCs. An algorithm to model check logic formulas in QCTL against a QMC model.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Some more notations
Let SO(H) be the set of super-operators on H, ranged over by E, F, · · · . Definition Let E , F ∈ SO(H). 1
2
E v F if for any ρ ∈ D(H), F (ρ) − E (ρ) is positive semi-definite; E . F if for any ρ ∈ D(H), tr(E (ρ)) ≤ tr(F (ρ)).
Let h be . ∩ &; it is obviously an equivalence relation.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Some notations
Let
SI(H) = {E ∈ SO(H) : E . IH } be the ‘quantum’ correspondence of the unit interval [0, 1] for real numbers.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Quantum Markov chains
A super-operator weighted Markov chain, or quantum Markov chain (QMC), over H is a tuple (S, Q, AP, L), where S is a countable set of states;
Q : S × S → SI(H) such that for each s ∈ S, ∑t ∈S Q(s, t ) h IH , AP is a finite set of atomic propositions; L is a mapping from S to 2AP . A classical Markov chain may be viewed as a degenerate quantum Markov chain in which all super-operators appear in the transition matrix have the form p IH for some 0 ≤ p ≤ 1.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Example: quantum loop
A simple quantum loop program goes as follows: l0 : q := F (q )
l1 : while M [q ] do
l2 : l3 : od where M = λ0 |0ih0| + λ1 |1ih1|.
q : = E (q )
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Example: quantum loop
l0 Fq Eq
l2
l1 Eq0 Eq1
I
Here Eq0 = {|0iq h0|} and Eq1 = {|1iq h1|}.
l3
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
QCTL
The syntax of quantum computation tree logic (QCTL) is as follows: Φ ::=
a | ¬Φ | Φ ∧ Ψ | Q∼E [ψ]
ψ ::= XΦ | ΦUΨ
where a is an atomic proposition, ∼ ∈ {., &}, and E ∈ SI(H). We call Φ a state formula and ψ a path formula.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
QCTL Let M = (S, Q, AP, L). The satisfaction relation |= is defined inductively: for any state s ∈ S, s |= a iff a ∈ L(s )
s |= ¬Φ iff s 6|= Φ
s |= Φ ∧ Ψ iff s |= Φ and s |= Ψ and for any path π ∈ PathM (s ), π |= XΦ
π |= ΦUΨ
iff iff
π (1) |= Φ
∃i ∈ N.(π (i ) |= Ψ ∧ ∀j < i.(π (j ) |= Φ)).
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
QCTL
Finally, s |= Q∼E [ψ] iff Q M (s, ψ) ∼ E where Q M (s, ψ) = Qs ({π ∈ PathM (s ) | π |= ψ}). But how to define Qs ?
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Super-operator valued measures
Let (Ω, Σ) be a measurable space; that is, Ω is a non-empty set and Σ a σ-algebra over Ω. A function ∆ : Σ → SI(H) is said to be a super-operator valued measure (SVM for short) if ∆ satisfies the following properties: 1 2
∆(Ω) h IH ;
∆( i Ai ) h ∑i ∆(Ai ) for all pairwise disjoint and countable sequence A1 , A2 , . . . in Ω. U
We call the triple (Ω, Σ, ∆) a (super-operator valued) measure space.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Properties of super-operator valued measures Let (Ω, Σ, ∆) be a measure space. Then 1 2 3 4
∆ ( ∅ ) = 0H ;
∆(Ac ) + ∆(A) h IH ;
for any A, A0 ∈ Σ, if A ⊆ A0 then ∆(A) . ∆(A0 ); for any sequence A1 , A2 , . . . in Σ, if A1 ⊆ A2 ⊆ . . . , then there exists a sequence E1 v E2 v . . . in SI(H) such that for any i, S ∆(Ai ) h Ei , and ∆( i ≥1 Ai ) = limi →∞ Ei . if A1 ⊇ A2 ⊇ . . . , then there exists a sequence E1 w E2 w . . . in SI(H) such that for any i, T ∆(Ai ) h Ei , and ∆( i ≥1 Ai ) = limi →∞ Ei .
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
SVM for a QMC Fix a state s ∈ S.
Sample space Ω = PathM (s ).
b ) ⊆ PathM (s ) be defined Let the cylinder set Cyl (π as b ) = {π ∈ PathM (s ) : π b is a prefix of π }; Cyl (π that is, the set of all infinite paths with b prefix π. σ-algebra over Ω: M b) : π b ∈ Pathfin Σs = σ ({Cyl (π (s )}
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
SVM for QMCs
M (s ), we b = s0 . . . sn ∈ Pathfin For any finite path π define the super-operator IH , if n = 0; b) = Q( π Q(sn−1 , sn ) · · · Q(s0 , s1 ), otherwise.
Let a mapping Qs be defined by letting Qs (∅) = 0H and b )) = Q(π b ). Qs (Cyl (π (1)
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Extend Qs to a SVM
Theorem The mapping Qs can be extended to a SVM on the σ-algebra Σs . Furthermore, this extension is unique up to the equivalence relation h. Remark: The main tool we use to prove this theorem is the Kluvanek’s generalisation of the Carath´eodory-Hahn extension theorem from vector measure theory.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
QCTL
Theorem For each path formula ψ and each state s in a QMC M, the set
{π ∈ PathM (s ) | π |= ψ} is measurable.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Back to the example l0 Fq Eq
l2
l1 Eq0 Eq1
I
l3
Let ♦Ψ ≡ ttUΨ. The QCTL formula Q&E [♦ l3 ] asserts that the probability that the loop program terminates is lower bounded by E . That is, for any initial quantum state ρ, the termination probability is not less than tr(E (ρ)). In particular, the property that it terminates everywhere can be described as Q&IH [♦ l3 ].
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Model checking
Given a state s in a qMC M = (S, Q, AP, L) and a state formula Φ expressed in QCTL, model checking if s |= Φ is essentially to determine whether s belongs to the satisfaction set Sat (Φ) = {s ∈ S : s |= Φ} which is defined inductively as follows: Sat (a) = {s ∈ S : a ∈ L(s )}
Sat (¬Ψ) = S \Sat (Ψ)
Sat (Ψ ∧ Φ) = Sat (Ψ) ∩ Sat (Φ)
Sat (Q∼E [ψ]) = {s ∈ S : Q M (s, ψ) ∼ E }. Recall:
Q M (s, ψ) = Qs ({π ∈ PathM (s ) | π |= ψ})
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Case 1: ψ = XΦ
By definition, {π ∈ PathM (s ) : π |= XΦ} = Thus Q M (s, XΦ) = Qs
=
∑
]
Cyl (st ) h
t ∈Sat (Φ)
U
t ∈Sat (Φ)
∑
Cyl (st ).
Qs (Cyl (st ))
t ∈Sat (Φ)
Q(s, t ).
t ∈Sat (Φ)
This can be calculated easily since by the recursive nature of the definition, we can assume that Sat (Φ) is already known.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Case 2: ψ = ΦUΨ
In this case, after some calculation, we get the I , H 0H , Q M (s, ΦUΨ) h ∑ Q M (t, ΦUΨ)Q(s, t ), t ∈S
equation system if s ∈ Sat (Ψ); if s 6∈ Sat (Φ) ∪ Sat (Ψ); if s ∈ Sat (Φ)\Sat (Ψ).
Then for each s ∈ Sat (Φ)\Sat (Ψ), Q M (s, ΦUΨ) h
∑
t ∈Sat (Φ)\Sat (Ψ)
Q M (t, ΦUΨ)Q(s, t ) +
∑
t ∈Sat (Ψ)
Q(s, t ).
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Let S 0 = Sat (Φ)\Sat (Ψ). For any s ∈ S 0 , Q M (s, ΦUΨ) h
∑0 Q M (t, ΦUΨ)Q(s, t ) + ∑
Q(s, t ).
t ∈Sat (Ψ)
t ∈S
Let
T = [Q(t, s )]s,t ∈S 0 and
"
G=
∑
# Q(s, t )
t ∈Sat (Ψ)
. s ∈S 0
Then the required row vector (Q M (s, ΦUΨ))s ∈S 0 is equivalent to the fixed point of the function f (X ) = X T + G .
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
A theorem
Theorem Let f (X ) = X T + G be defined above. Then 1
2
0
f (X ) has the least fixed point, denoted by E 0 , in SI(H)|S | under the order v; Given any E ∈ SI(H) and 1 ≤ i ≤ |S 0 |, it can be decided whether E ∼ Ei0 , ∼ ∈{., &}, in time O (n2 d 4 ) where d = dim (H) is the dimension of H and n = |S 0 |.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Back to the example again We check the property Q&E [♦ l3 ] = Q&E [ttUl3 ] when F = {|+ihi | : i = 0, 1}, E i = {|i ihi |}, i = 0, 1, and E = X . l0 Fq Eq
l2
l1 Eq0 Eq1
I
l3
We first calculate that Sat (l3 ) = {l3 } and Sat (tt) = {l0 , l1 , l2 , l3 }.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Back to the example again
l0 Fq Eq
l2
l1 Eq0 Eq1
I
l3
Q M (l0 , ♦ l3 ) = Q M (l1 , ♦ l3 )F Q M (l1 , ♦ l3 ) = Q M (l2 , ♦ l3 )E 1 + E 0 Q M (l2 , ♦ l3 ) = Q M (l1 , ♦ l3 )E
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Example
We calculate that for i = 0, 1, 2, Q M (li , ♦ l3 ) = Set 0 where Set 0 = {|0ih0|, |0ih1|} h I , and so li |= Q&E [♦ l3 ] for any E . I .
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Outline
1
Motivation
2
Basic notions from quantum information theory
3
Quantum Markov chain
4
Quantum computation tree logic
5
Algorithm
6
Summary
Algorithm
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Summary
Summary
A super-operator weighted Markov chain model which aims at providing finite models for general quantum programs and quantum communication protocols. A quantum extension QCTL of the logic PCTL to descibe properties we are interested in for QMCs. An algorithm to model check logic formulas in QCTL against a QMC model.
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Topics for further studies
Tools to implement the model checking algorithm. Model checking quantum properties. Check security of physically implemented quantum cryptographic systems.
Summary
Motivation
Basic notions from QIP
Quantum Markov chain
Quantum computation tree logic
Algorithm
Thank you! Questions or Comments?
Summary
